2003 AIME I Problems/Problem 5
Problem
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is where
and
are positive integers, and
and
are relatively prime, find
Solution
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The set can be broken into several parts: the big parallelepiped, the 6 external parallelepipeds that each share a face with the large parallelepiped, the th spheres (one centered at each vertex of the large parallelepiped), and the
th cylinders connecting each adjacent pair of spheres.
- The volume of the parallelepiped is
cubic units.
- The volume of the external parallelepipeds is
.
- There are 8
th spheres, each of radius
. Together, their volume is
.
- There are 12
th cylinders, so 3 complete cylinders can be formed. Their volumes are
,
, and
, adding up to
.
The combined volume of these parts is . Thus, the answer is
.
See also
2003 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |